arXiv subjects

David Helm

Publications and source records attributed to David Helm.

29 records · Page 2Linked to original sources

arXiv2012

Whittaker models and the integral Bernstein center for GL_n

We establish integral analogues of results of Bushnell and Henniart for spaces of Whittaker functions arising from the groups GL_n(F) for F a p-adic field. We apply the resulting theory to the existence of representations arising from the conjectural "local Langlands correspondence in families" of Emerton-Helm, and reduce the question of the existence of such representations to a natural conjecture relating the integral Bernstein center of GL_n(F) to the deformation theory of Galois representations.

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arXiv2011

The local Langlands correspondence for GL_n in families

Let E be a nonarchimedean local field with residue characteristic l, and suppose we have an n-dimensional representation of the absolute Galois group G_E of E over a reduced complete Noetherian local ring A with finite residue field k of characteristic p different from l. We consider the problem of associating to any such representation an admissible A[GL_n(E)]-module in a manner compatible with the local Langlands correspondence at characteristic zero points of Spec A. In particular we give a set of conditions that uniquely characterise such an A[GL_n(E)]-module if it exists, and show that such an A[GL_n(E)]-module always exists when A is the ring of integers of a finite extension of Q_p. We also use these results to define a "modified mod p local Langlands correspondence" that is more compatible with specialization of Galois representations than the mod p local Langlands correspondence of Vigneras.

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arXiv2011

Monodromy Filtrations and the Topology of Tropical Varieties

We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space Γ_X. We give a geometric interpretation of the cohomology of Γ_X in terms of the action of a monodromy operator on the cohomology of X. This gives bounds on the Betti numbers of $Γ_X$ in terms of the Betti numbers of $X$. When $X$ is a sufficiently general complete intersection, this allows us to show that the cohomology of Trop(X) vanishes in degree less than dim(X). In addition, we give a description for the top power of the monodromy operator acting on middle cohomology in terms of the volume pairing on $Γ_X$.

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arXiv2009

On $l$-adic families of cuspidal representations of $\GL_2(\Q_p)$

We compute the universal deformations of cuspidal representations $π$ of $\GL_2(F)$ over an algebraically closed field of characteristic $l$, where $F$ is a local field of residue characteristic $p$ not equal to $l$. When $π$ is supercuspidal there is an irreducible, two-dimensional representation $ρ$ of $G_F$ that corresponds to $π$ by the mod $l$ local Langlands correspondence of Vign{é}ras; we show there is a natural isomorphism between the universal deformation rings of $π$ and $ρ$ that induces the usual local Langlands correspondence on characteristic zero points. Our work establishes certain cases of a conjecture of Emerton.

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arXiv2009

Towards a Jacquet-Langlands correspondence for unitary Shimura varieties

Let G be a unitary group over a totally real field, and X a Shimura variety for G. For certain primes p of good reduction for X, we construct cycles on the characteristic p fiber of X. These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on X. The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group G', which is isomorphic to G at finite places but not isomorphic to G at archimedean places. More precisely, each such cycle has a natural desingularization, and this desingularization is "almost" isomorphic to a scheme parametrizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety X' arising from G'. We exploit this relationship to construct an injection of the etale cohomology of X' into that of X. This yields a geometric construction of "Jacquet-Langlands transfers" of automorphic representations of G' to automorphic representations of G.

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arXiv2009

A geometric Jacquet-Langlands correspondence for U(2) Shimura varieties

Let G be a unitary group over the rationals, associated to a CM-field F with totally real part F^+, with signature (1,1) at all the archimedean places of F^+. Under certain hypotheses on F^+, we show that Jacquet-Langlands correspondences between certain automorphic representations of G and representations of a group G' isomorphic to G except at infinity can be realized in the cohomology of Shimura varieties attached to G and G'. We obtain these Jacquet-Langlands correspondences by studying the bad reduction of a Shimura variety X attached to G at a prime p for which X has maximal parabolic level structure. We construct a "Deligne-Rapoport" model for X and show that the irreducible components of its special fiber have a global structure that can be explicitly described in terms of Shimura varieties X' for unitary groups G' isomorphic to G except at infinity. The weight spectral sequence of Rapoport-Zink then yields an expression for certain pieces of the weight filtration on the etale cohomology of X in terms of the cohomology of a suitable X'. This identifies a piece of this weight filtration with a space of algebraic modular forms for G'. A consequence is certain cases of the Jacquet-Langlands correspondence between G and G' in terms of a canonical isomorphism between spaces of arithmetic interest, rather than simply as an abstract bijection between isomorphism classes of representations.

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arXiv2006

Mazur's principle for U(2,1) Shimura varieties

Mazur's principle gives a criterion under which an irreducible mod l Galois representation arising from a classical modular form of level Np (with p prime to N) also arises from a classical modular form of level N. We consider the analogous question for Galois representations arising from certain unitary Shimura varieties. In particular, we prove an analoge of Mazur's principle for U(2,1) Shimura varities. We also give a conjectural criterion for a Galois representation arising in the cohomology of a unitary Shimura variety whose level subgroup is parahoric at p to also arise in the cohomology of a Shimura variety with "less level structure at p".

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arXiv2006

Flatness of the linked Grassmannian

We show that the linked Grassmannian scheme, which arises in a functorial compactification of spaces of limit linear series, and in local models of certain Shimura varieties, is Cohen-Macaulay, reduced, and flat. We give an application to spaces of limit linear series.

math.AG↗

arXiv2004

On maps between modular Jacobians and Jacobians of Shimura curves

By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it is known that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve of discriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-new subvariety of J_0^D(pqN), whenever p and q are distinct primes not dividing D or N. However, this approach is entirely nonconstructive, and gives no information at all about any particular isogeny between the two varieties. In this paper, we determine the kernels of all possible maps from one variety to the other up to support on a small finite set of maximal ideals of the Hecke algebra. We also examine consequences of this computation for the geometry of Jacobians of Shimura curves. In particular, we obtain a "multiplicity one" result for certain Galois representations in the torsion of Jacobians of such curves.

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arXiv2003

Algorithms for graded injective resolutions and local cohomology over semigroup rings

Let Q be an affine semigroup generating Z^d, and fix a finitely generated Z^d-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z^d-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules H^i_I(M) supported on any monomial (that is, Z^d-graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them.

math.AC↗

arXiv2000

Bass Numbers of Semigroup-Graded Local Cohomology

Given a module M over a ring R which has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology of M at any Q-graded ideal I in terms of Ext modules. This method is used to obtain finiteness results for the local cohomology of graded modules over semigroup rings; in particular we prove that for a semigroup Q whose saturation is simplicial, the Bass numbers of such local cohomology modules are finite. Conversely, if the saturation of Q is not simplicial, one can find a graded ideal I and a graded R-module M whose local cohomology at I in some degree has an infinite-dimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup.

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